East Asian Journal on Applied Mathematics
The East Asian Journal on Applied Mathematics (EAJAM) aims at promoting study and research in Applied Mathematics in East Asia. It is the editorial policy of EAJAM to accept refereed papers in all active areas of Applied Mathematics and related Mathematical Sciences. Novel applications of Mathematics in real situations are especially welcome. Substantial survey papers on topics of exceptional interest will also be published occasionally.
Energy-Stable Full Discretization of the Modified Elastic Flow of Closed Curves
An energy-stable full discretization for the modified elastic flow of closed curves is proposed. This is a gradient flow of a modified elastic energy combining bending and Dirichlet energies. The minimization of Dirichlet energy can lead to improved mesh quality. Gradient flows for both isotropic and anisotropic cases are considered. We derive new evolution equations for the parameterization and curvature vector of curves in arbitrary codimension. The proposed formulation is discretized by a parametric finite element method in space and a first-order implicit scheme in time. We establish the unconditional energy stability for the fully discretized scheme. Additionally, the second-order accuracy of the BDF2 scheme is demonstrated. Numerical examples in two and three dimensions illustrate the efficiency, energy stability, and asymptotic mesh distribution of the method for simulating the modified elastic flow.
Copyright (c) 2025 East Asian Journal on Applied Mathematics
Quasi-Periodic Breathers of the Hirota-Maccari System
The Hirota-Maccari (HM) system generalizing the Hirota equation, serves as a generalized (2+1)-dimensional model in fluid dynamics, plasma physics, and optical fiber communication. In this paper, we obtain the quasi-periodic breathers to HM system by use of Hirota’s bilinear method and the theta function. Asymptotic analysis demonstrates that the quasi-periodic breathers can be reduced to regular breathers under small amplitude limits. Moreover, we also classified solutions based on their asymptotic behavior. Numerical examples are given to confirm the theoretical analysis.
High-Order BDFk Parametric Finite Element Methods for Anisotropic Surface Diffusion Flows and Applications in Solid-State Dewetting
In this paper, we extend the BGN formulation [J.W. Barrett, H. Garcke and R. Nürnberg, J. Comput. Phys. 222 (2007)] by incorporating the $k$-order backward differentiation formulae (BDFk) for time discretization. This allows us to develop high-order temporal parametric finite element methods for simulating anisotropic surface diffusion flows and solid-state dewetting problems, achieving accuracy levels from second-order to fourth-order. We prove the well-posedness of the constructed high-order schemes. The proposed schemes maintain good mesh quality characteristic of the classical first-order BGN scheme. Finally, we present several numerical simulations to demonstrate the high-order temporal accuracy and verify the preservation of good mesh quality and energy stability throughout the evolution.
A Greedy Randomized Average Block Projection Method for Linear Feasibility Problems
The randomized projection (RP) method is a simple iterative scheme for solving linear feasibility problems and has gained popularity due to its speed and low memory requirement. This paper develops an accelerated variant of the standard RP method by using two ingredients: the greedy probability criterion and the average block approach, and obtains a greedy randomized average block projection (GRABP) method for solving large-scale systems of linear inequalities. We demonstrate that the GRABP method achieves deterministic linear convergence with various extrapolated step sizes. Numerical experiments on both randomly generated and real-world data show the advantage of GRABP over several state-of-the-art solvers, such as the RP method, the sampling Kaczmarz Motzkin (SKM) method, the generalized SKM method, and the Nesterov acceleration of SKM method.
An Energy Stable TFPM-Based Petrov-Galerkin Scheme for Solving the Allen-Cahn Equation
An energy stable tailored finite point method based Petrov-Galerkin scheme to solve Allen-Cahn equation is proposed. In time discretization, we present both first-order and second-order semi-discrete schemes based on stabilized and convex-splitting techniques, which satisfy unconditional energy stability. We prove the maximum bound preserving principle for first-order schemes. Due to nonlinearity, the well-posedness of weak formulations based on semi-discrete schemes are demonstrated. As the nature of singularly perturbation in semi-discrete level remains when $ε$ is extremely small, we establish a specified Petrov-Galerkin scheme which leads to a unified way for space discretization. To this end, we set up nonlinear solvers which are proved to be stable and convergent. Then we construct our Petrov-Galerkin scheme, which is built upon problem-dependent test function space. The stability and second-order convergence of this scheme are rigorously proved in one dimension. In order to compute test functions, specialized TFPM schemes are incorporated into the scheme. Numerical experiments show the accuracy, efficiency, and the good performance of the method on uniform meshes even when mesh size $h$ is much larger than $ε$.
Piecewise Smooth $N$-Dimensional Nonlinear Singular Singularly Perturbed Boundary Value Problems
Internal layer phenomena are often appear in various piecewise smooth nonlinear singular singularly perturbed problems of natural sciences. To characterize these special structures, we study piecewise smooth $n$-dimensional nonlinear singular singularly perturbed boundary value problems. In particular, we show the existence of solutions with an internal layer and construct their asymptotic expansions. The remainder estimations of the approximate solutions are also given. Finally, an example aimed to verify the correctness of the developed theory is presented.
Schwartz Duality for Singularly Perturbed Differential Equations with Chebyshev Spectral Methods
Singularly perturbed differential equations with the Dirac delta function usually yield discontinuous solutions. Therefore, careful consideration is required when using numerical methods to solve these equations because of the Gibbs phenomenon. A remedy based on the Schwartz duality has been previously proposed, yielding superior results without oscillations. However, this approach has primarily been applied to linear problems and still exhibits the Gibbs phenomenon when extended to nonlinear or higher-dimensional problems. In this paper, we propose a consistent yet simple approach based on Schwartz duality that can handle such problems. Our proposed approach utilizes a modified direct projection method with a consistent discrete derivative of the Heaviside function, which directly approximates the Dirac delta function. As numerical examples, we consider several problems, including the Burgers’ equation and the two-dimensional time-dependent advection equation. The proposed method effectively eliminates Gibbs oscillations without the need for traditional regularization and demonstrates uniform error reduction for the problems considered.
Differential equations with singular sources
A New Multilevel Homotopic Adaptive Finite Element Algorithm for Convection-Dominated Diffusion-Reaction Problems
The multilevel homotopic adaptive finite element method (MHAFEM) was developed for solving convection-dominated diffusion-reaction problems on triangular meshes, where the mesh adaptation (refinement and mesh smoothing/moving) is based on an optimal interpolation error estimate in $L^p$ norm. In this paper, we follow the framework of the MHAFEM to give a new multilevel homotopic adaptive finite element algorithm. This new MHAFEM executes mesh refinement based on an a posteriori error estimator instead of the interpolation error estimate. We apply this algorithm to 1D and 2D example problems using 1D linear elements and 2D quadrilateral transition elements, respectively. Numerical results demonstrate the remarkable efficiency of the algorithm.
Convection-dominated diffusion-reaction problem
multilevel homotopic adaptive finite element method
A Fully Decoupled Linear Second-Order Energy Stable Numerical Scheme for the Moving Contact Line Problem
A decoupled linear second-order accurate unconditionally energy stable scheme for a phase-field model of the moving contact line (MCL) problem is proposed. The model consists of coupled Cahn-Hilliard and Navier-Stokes equations with the generalized Navier boundary condition. The scheme proposed introduces three scalar auxiliary variables to decouple nonlinear coupling terms while retaining stability and accuracy. Two of these variables represent the bulk and boundary energy, and one captures the “zero-energy-contribution” property between convection and surface tension terms. All related terms are decoupled by applying semi-explicit treatments while preserving stability and accuracy. Further coupling between velocity and pressure is removed by adopting a projection method. The overall scheme is second-order accurate, and the unconditional stability of energy is rigorously proved. Numerical results demonstrate the accuracy, efficiency, and stability of the method for the MCL problem. Besides, a consistent implementation of the contact angle hysteresis (CAH) is conducted to model important CAH phenomena.
The Quasi-Reversibility Regularization for Source Inversion of a Linear Kuramoto-Sivashinsky Equation
The source inversion problem for the linear Kuramoto-Sivashinsky equation is studied. To address the ill-posedness of the source term inversion, a quasi-reversibility regularization method is used for the stable reconstruction of the source term. In the case of noisy measurement data, the convergence (error estimate) of regularized solutions with respect to noise level is analyzed under a priori choice of the regularization parameter. Subsequently, a modified Morozov discrepancy principle is proposed to select regularization parameters for the a posteriori principle. The convergence (error estimate) of regularized solutions is established under this a posteriori principle. Numerical examples are provided to verify the effectiveness of the proposed quasi-reversibility regularization method.
A Fast Cascadic Multigrid Method for Exponential Compact FD Discretization of Singularly Perturbed Convection-Diffusion Equations
Solving singularly perturbed convection-diffusion equations, especially for 3D problems, is a challenging problem. In this paper, we extend our work on the extrapolation cascadic multigrid (EXCMG) method for solving the 3D Poisson equation — cf. [Pan et al., J. Sci. Comput. 2017], to 3D convection-diffusion equations with singularly perturbed parameters. First, we introduce an exponential higher order compact finite difference scheme to discretize the 3D convection-diffusion equation with variable convection coefficients, resulting in a larger-scale nonsymmetric linear system. Then, we propose an EXCMG method combined with the biconjugate gradient stabilized smoother to solve the larger-scale nonsymmetric system efficiently. Numerical experiments demonstrate that the EXCMG method is a highly effective solver for convection-dominated problems, which outperforms the existing multigrid methods such as aggregation-based algebraic multigrid method.
Geometric Approach to Symmetric Positive Definite Linear Systems
This paper compares the performance of the conjugate gradient method and geometric approach in the case of symmetric positive definite (SPD) linear systems. This approach is based on the geometric theory of ODEs which was effectively initiated by Poncaré and Liapunov. The simplest and most obvious advantage of the geometric approach over the conjugate gradient method (the MATLAB code pcg) is that this approach can find the inverse of the underlying positive definite matrix and the solution. We present various numerical examples, which demonstrate the superiority of the geometric approach. For SPD linear systems, this approach provides much higher accuracy than the conjugate gradient method. In particular, since it is a one-stop procedure, it can avoid the growth of accumulated round-off errors to some extent.
Identifying the Order and a Space Source Term in a Time Fractional Diffusion-Wave Equation
This paper is devoted to identifying the order of time fractional derivative and a space-dependent source term in a time fractional diffusion-wave equation from some additional measured data in a subdomain or on a subboundary with a small time period. The Lipschitz continuity of forward operators mapping the unknown order and source term into the given data are established based on the stability estimates of solution for the direct problem. We prove the uniqueness of the considered inverse problems by using the asymptotic behavior of the solution at $t$ = 0, the Titchmarsh convolution theorem and the Duhamel principle. Moreover, a Tikhonov-type regularization method is proposed with $H^1$-norm as a penalty term. The existence of the regularized solution and its convergence to the exact solution under a suitable regularization parameter choice are obtained. Then we employ a linearized iteration algorithm combined with the piecewise linear finite element approximation to find simultaneously the approximate order and space source term. Three numerical examples for one- and two-dimensional cases are tested and the numerical results demonstrate the effectiveness of the proposed method.
Asymptotic Behavior of Solutions for a Free Boundary Problem with a Monostable Nonlinearity
A free boundary problem with Dirichlet boundary conditions is studied. Such problems can be used for describing the spread of chemical substances or biological species, which live in a moving region $[0,h(t)].$ In this case, the free boundary $h(t)$ represents the spreading front. If the density of the substance or the population at the boundary exceeds a threshold value, they will be going to spread outwards. On the other hand, the outside environment may be not very beneficial for spreading and this generates a decay rate. We mainly analyze how the decay rate and threshold value affect the solutions. There is a trichotomy result: the solution is either shrinking, or the transition case, or spreading. Besides, if the decay rate or threshold value is large, only the shrinking happens.
High-Order Operator-Compensation Schemes with Mass and Energy Conservation for the Multi-Dimensional Zakharov System
This paper presents high-order operator-compensation (OC) methods and the Crank-Nicolson (CN) technology to solve the Zakharov system (ZS). Firstly, the OC method is used for spatial discretization, and semi-discrete high-order OC schemes are obtained, which can maintain mass and energy conservation. Subsequently, the CN technology is adopted for temporal discretization, and full-discrete high-order OC schemes are obtained, which can inherit mass and energy conservation and achieve arbitrary even-order accuracy in space and second-order accuracy in time. In the process of solving ZS, we employ the linearization method to handle the nonlinear ZS, which significantly improves the computational efficiency. Considering the large stencil of numerical schemes, we propose a fast numerical iterative method to deal with the problem, which saves the computational cost. Numerical experiments are given to test accuracy order, verify conservation property, and simulate dynamic evolution.
Exact Solutions for a Higher-Order Boussinesq Equation and a New Higher-Order Boussinesq-Like Equation
A kind of higher-order Boussinesq equation is studied in this work. Based on the Bell polynomials theories, bilinear representations of the equation are derived, and diverse interaction solutions are constructed through Hirota’s bilinear method. These include interaction solutions characterized by hyperbolic cosine and cosine functions, as well as interactions between lump and soliton solutions. In addition, generalized bilinear operators are used in order to construct a new higher-order Boussinesq-like equation, while lump and breather solutions are also developed utilizing Hirota’s bilinear technique. For the various explicit solutions obtained in this work, several of them are considered to selected particular values for the relevant parameters in order to plot different kind of three-dimensional surfaces with associated two-dimensional density profiles to give a comprehensive understanding of the evolution for various solutions.
Improved Random Feature Method for Continuous Solution Reconstruction from Sparse Observation
The reconstruction of continuous solutions using all available mechanisms and data is essential for high-precision simulations and forecasts. This paper presents an improved random feature method (IRFM) that combines observational data with models for solution reconstruction. For the multi-region neuron approximation, we employ a global activation function with robust local approximation capabilities instead of the traditional piecewise method. This enhances smoothness across regions and accelerates convergence. We also derived the equivalence condition for the optimal solution of multi-constrained optimization problems, established criteria for determining the weights in the cost function and the distribution of randomly generated collocation points, reducing biases from subjective choices. Additionally, we introduce a weighted scheme for computing the cost function related to sparse observations, reducing interpolation errors and improving stability against noise. Numerical examples demonstrate that the IRFM is more accurate and converges faster than the original RFM. Its efficiency and accuracy are validated through comparisons with physics-informed neural networks, and its flexibility is shown by successful continuous solution reconstruction in complex domains.
A Fast Discontinuous Galerkin Finite Element Method for a Bond-Based Linear Peridynamic Model
A fast discontinuous Galerkin finite element algorithm on a non-uniform mesh for solution of a one-dimensional bond-based linear peridynamic model with fractional noise is developed. It is based on the approximation of the stiffness matrix corresponding to the discontinuous Galerkin finite element method by its hierarchical representation. The fast algorithm reduces the storage requirement for the stiffness matrix from $\mathscr{O} (N^2 )$ to $\mathscr{O} (kN),$ where $k$ is a parameter controlling the accuracy of hierarchical matrices. The computational complexities of assembling the stiffness matrix and the Krylov subspace method for solving linear systems are also reduced from $\mathscr{O} (N^2)$ to $\mathscr{O} (kN).$ Numerical results show the utility of the numerical method.
Error Analysis of $hp$ Spectral Element Approximation for Optimal Control Problems with Control Constraint
In this paper, an $hp$ spectral element approximation for distributed optimal control problem governed by an elliptic equation is investigated, whose objective functional does not include the control variable. And the constraint set on control variable is stated with $L^2$-norm. Optimality condition of the continuous and discretized systems are deduced. In order to solve the equivalent systems with high accuracy, $hp$ spectral element method is employed to discretize the constrained optimal control systems. Based on the property of some interpolation operators, a posteriori error estimates are also established by using some properties of some interpolation operators carefully. Finally, a projection gradient algorithm and a numerical example are provided, which confirm our analytical results. Such estimators guarantee the construction of reliable adaptive methods for optimal control problems.
Multi-View Data Clustering via Dynamical Optimization of Consensus Laplacian Matrix
Multi-view data analysis has gained increasing popularity, in particular multi-view spectral clustering has attracted much attention for its outstanding performance in mining heterogeneity information in multi-view data. However, most spectral clustering methods exhibit the following disadvantages: firstly, learning consensus representation directly from multi-view data that may contain noise renders a distorted description; secondly, the traditional two-step process may fall into a suboptimal solution. To overcome these disadvantages, a novel multi-view spectral clustering method is proposed by unifying the optimization of consensus Laplacian matrix and the learning and discretization of spectral embedding into one step. We consider that the optimal Laplacian matrix is in the neighborhood of view-specific Laplacian matrix, as the view-specific Laplacian matrix only contains partial information from multi-view data, resulting in certain deviation from the optimal Laplacian matrix. The consensus Laplacian matrix was obtained in a dynamic optimization way with the spectral rotation and embedding information simultaneously determined. Extensive experiments have been conducted to demonstrate the effectiveness and superiority of our proposed method.
Higher-Order Rogue Waves of the Nonlocal Nonlinear Schrödinger Equation in the Defocusing Regime
The previous studies have shown that the defocusing NLSE has no the modulational instability, and was not found to admit the rogue wave phenomenon so far. In this paper, we address the question of the higher-order rogue wave solutions of the nonlocal $\mathscr{P}\mathscr{T}$-symmetric NLSE in the defocusing regime. Based on Darboux transformation and iterations, we derive an explicit solution for the higher-order rogue waves by adopting a variable separation and Taylor expansion technique. The higher-order rogue wave solutions are expressed in separation-of-variables form. Furthermore, in order to understand these solutions better, patterns of the rogue waves for lowest three order are explored clearly and conveniently. The reported results may be useful for the design of experiments for observation of rogue waves in the defocusing nonlinear physical systems.
Defocusing nonlocal nonlinear Schrödinger equation
Energy Stable Splitting Schemes for Maxwell’s Equations in Lorentz Media
In this paper, we introduce energy-stable schemes based on operator splitting methods for Maxwell’s equations in two-dimensional Lorentz dispersive media with transverse electric polarization, namely the sequential splitting scheme (SS-ML) and the Strang-Marchuk splitting scheme (SM-ML). Each splitting scheme involves two substages per time step, where 1D discrete sub-problems are solved using the Crank-Nicolson method for time discretization. Both schemes ensure energy decay and unconditional stability. The convergence analysis reveals that the SS-ML scheme exhibits first-order accuracy in time and second-order accuracy in space based on the energy technique, while the SM-ML scheme achieves second-order accuracy in both time and space. Additionally, numerical dispersion analysis yields two discrete numerical dispersion relation identities for each scheme. Theoretical results are supported by examples and numerical experiments.
The Shock Formation for 2D Isentropic Compressible Euler Equations with Damping
In this paper, we investigate the blowup mechanism for the 2D isentropic compressible Euler equations with a damping term. Generally, damped Euler equations have a global classical solution for small smooth initial data. However, we identify a class of large initial data for the damped Euler equations, under the condition of azimuthal symmetry and by choosing a suitable self-similar transformation, resulting in the solution to the Cauchy problem blowing up in a finite time. Additionally, we introduce modulation variables to accurately describe the blowup time and location. Furthermore, the blowup profile exhibits a cusp singularity with Hölder $C^{1/3}$ regularity at the blowup point.
Crank-Nicolson ADI Finite Difference Method for Three-Dimensional Nonlinear Partial Integro-Differential Equations with Weak Singular Kernels
The main objective of this study is to present a fast and efficient numerical scheme for solving nonlinear integral differential equations with weak singular kernels in three-dimensional domain. First, the temporal derivative and integral term are approximated by the Crank-Nicolson (CN) method and the second-order fractional quadrature rule. After that the spatial discretization is carried out by combining the finite difference method and the alternating direction implicit (ADI) method, and the nonlinear term are approximated using the Taylor expansion. The stability and convergence of the proposed scheme are analyzed, followed by the verification of the theoretical results through numerical experiments.
Three-dimensional nonlinear integral differential equation
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A Generalized Quasi-Boundary Value Method for an Inverse Source Problem in a Distributed Order Time-Fractional Diffusion Equation
Subdiffusion equations with distributed-order fractional derivatives describe important physical phenomena. In this paper, we consider an inverse space-dependent source term problem governed by a distributed order time-fractional diffusion equation using final time data. Based on the series expression of the solution, the inverse source problem can be transformed into a first kind of Fredholm integral equation. The existence, uniqueness and a conditional stability of the considered inverse problem are established. Building upon this foundation, a generalized quasi-boundary value regularization method is proposed to solve the inverse source problem, and then we prove the well-posedness of the regularized problem. Further, we provide the convergence rates of the regularized solution by employing an a priori and an a posteriori regularization parameter choice rule. Numerical examples in one-dimensional and two-dimensional cases are provided to validate the effectiveness of the proposed method.
distributed order time-fractional diffusion equation
Local Structure Preserving Algorithms for Molecular Beam Epitaxial Models with Slope Selection Based on Energy Quadratization Method
A molecular beam epitaxial (MBE) model with slope selection consisting of a fourth order Ginzburg-Landau double well potential, is derived from the variation of the free energy. One challenge in constructing the local structure preserving algorithms (SPAs) for the MBE model with slope selection is how to properly discretize the equation in space and time simultaneously in order to preserve the local structure at the discrete level. To resolve this issue, we employ the local energy dissipation property and the energy quadratization techniques. One novelty is that all nonlinear terms are treated semi-explicitly. The other novelty is that we introduce proper intermediate variables to make the space operators act on one single term, which is one crucial step in constructing local SPAs. We then develop two local energy dissipation preserving schemes and show rigorously the local energy dissipation property of the two schemes. Under suitable boundary conditions, such as periodic boundary conditions, the algorithms can preserve not only mass but also global energy dissipation property. Numerical experiments confirm the second-order accuracy and show the excellent performance of the schemes proposed.
Convergence of a Discontinuous Galerkin Method on Bakhvalov-Type Meshes for Singularly Perturbed Volterra Integro-Differential Equations
A discontinuous Galerkin (DG) method on Bakhvalov-type (B-type) meshes for singularly perturbed Volterra integro-differential equations (SPVIDEs) is proposed. We derive abstract error bounds of the DG method for the SPVIDEs in the $L^2$-norm. It is shown that the approximate solution generated by the DG method on B-type meshes has optimal convergence rate $k + 1$ in the $L^2$-norm, when using the piecewise polynomial space of degree $k.$ Numerical simulations demonstrate the validity of the theoretical results.
The Optimal Production Transport: Model and Algorithm
We propose the optimal production transport model, which is an extension of the classical optimal transport model. We observe in economics, the production of the factories can always be adjusted within a certain range, while the classical optimal transport does not take this situation into account. Therefore, differing from the classical optimal transport, one of the marginals is allowed to vary within a certain range in our proposed model. To address this, we introduce a multiple relaxation optimal production transport model and propose the generalized alternating Sinkhorn algorithms, inspired by the Sinkhorn algorithm and the double regularization method. By incorporating multiple relaxation variables and multiple regularization terms, the inequality and capacity constraints in the optimal production transport model are naturally satisfied. Alternating iteration algorithms are derived based on the duality of the regularized model. We also provide a theoretical analysis to guarantee the convergence of our proposed algorithms. Numerical results indicate significant advantages in terms of accuracy and efficiency. Furthermore, we apply the optimal production transport model to the coal production and transport problem. Numerical simulation demonstrates that our proposed model can save the production and transport cost by 13.17%.
generalized alternating Sinkhorn algorithm
Stomata are important for plants as nodes linking the transpiration process and photosynthesis process. Accurate regulation of stomatal aperture balances water loss and carbon-dioxide uptake. In a recent work, a mathematical model is developed for seed-plant stomata, which exhibits rich dynamical behaviors and can semi-quantitatively explain responses of the stomata to environmental changes. In this work, we numerically study the dynamical behaviors of the stomatal model using phase diagrams with respect to a few important parameters. The phase diagrams are helpful for further validation of the model and experimental measurements of model parameters.